Interacting random walk in a dynamical random environment. II. Environment from the point of view of the particle

A NNALES DE L ’I. H. P., SECTION B

C ARLO B OLDRIGHINI

R OBERT A. M INLOS

A LESSANDRO P ELLEGRINOTTI Interacting random walk in a dynamical random environment. II. Environment from the point of view of the particle

Annales de l’I. H. P., section B, tome 30, n

^o

4 (1994), p. 559-605

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Interacting random walk

in

a

dynamical random environment

II. Environment from the point ^{of view of the} particle

Carlo BOLDRIGHINI

(*)

Robert A. MINLOS

(*)

Alessandro PELLEGRINOTTI

(*)

Dipartimento ^di Matematica e Fisica, Universita di Camerino, Via Madonna delle Carceri, ⁶²⁰³² Camerino, Italy.

Moscow State University, Moscow, Russia.

Dipartimento ^di Matematica, Universita di Roma La Sapienza,

P. A. Moro 2, 00187 Roma, Italy.

Vol. 30, ^n° 4, 1994, p. 559-605. Probabilités et Statistiques

ABSTRACT. - We

consider,

^{as in}

I,

^a random walk

Xt

^{E E}

Z+

and a

dynamical

^random

field ~ (x),

^{x E Z"} in mutual interaction with each other. The model is a

perturbation

^{of un}

unperturbed

model in which walk and field evolve

independentently.

^{Here we} consider the environment process in ^a frame of reference that moves with the

walk, i.e.,

^{the "field} from the

point

^{of view of the}

particle" ~t (.) == çt (Xt +.).

We prove that its distribution

tends,

^{as t -~} oo, to ^a

limiting

distribution _tc, which is

absolutely

continuous with

respect

^{to the}

unperturbed equilibrium

distribution. We also prove

that, for v &#x3E; ^3,

the time correlations of the field

decay

^as

e-cxt

const

20142014.

^.

Key words: Random walk in random environment, ^mutual influence, environment from the

point of view of the particle.

(*) Partially supported by ^C.N.R. (G.N.F.M.) and M.U.R.S.T. research funds.

A.M.S. Classification : ^{60 J} 15, 60 J 10.

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques - ^0246-0203

Vol. 30/94/04/$ 4.00/@ Gauthier-Villars

560 C. BOLDRIGHINI, ^{R. A.} MINLOS AND A. PELLEGRINOTTI

On considere un chemin aleatoire

Xt

^{E E}

Z+

^et

un milieu aleatoire

dynamique çt (x), x

^{E Z" en} interaction mutuelle.

Le modele est une

perturbation

d’un modele

imperturbe,

^dans

lequel

le chemin aleatoire et le milieu evoluent d’une

façon independante.

^On

etudie ici le

proces

^{du milieu dans un}

repere qui

^se

deplace

^{avec le}

chemin

aleatoire,

c’ est-a-dire le ^« milieu du

point

^{de vue de la}

particule »

r~t

( . ) (Xt +.).

^{On montre} que la distribution

de T/t tend, pour t

^{-~ o0}

a une distribution limite

qui

^est absolument continue par

rapport

^a la distribution

d’ équilibre

^{du milieu}

imperturbe.

^{On montre} aussi que le

premier

^{terme du}

developpement asymptotique

^des correlations

temporelles

du

champ ~t

^{est donne} par ^const

t2 " e-at

^.

1. INTRODUCTION

The

present

paper is second in ^a series of two papers. As in the

preceding

paper

[1] (hereafter

^{referred to as} Part

I),

^we

study

the time evolution of a

random walk

Xt

^{on the} v-dimensional lattice Z" and a field

(environment) çt (x) :

^{x E}

subject

^{to a mutual} interaction of local character.

Time is

discrete, t

^E

Z+,

^{and the field E Z"} takes values in

a finite set S.

We

briefly

recall the main features of the

model,

and refer the reader to Part I for more details. The

assumptions

^{of the}

present

paper which differ from those of Part I are listed in Section 2.

As a

starting point

^{we consider an}

"unperturbed" model,

in which the random walk and the environment evolve

independently.

The random walk is

homogeneous

with transition

probabilities ~ Po (y) :

^{y E}

7~v ~,

^{and the}

evolution of the environment at each site is an

ergodic

^Markov

chain,

^{with a}

finite space ^state

S,

and stochastic

operator Qo = ~ qo (s, s’) : s,

^{s’ E}

S ~,

the same for all sites. _7ro will denote the

unique stationary

^{measure of the}

chain. The evolution at different sites is

independent.

For the

interacting

model the random walk transition

probabilities

^are

written as

Annales de l ’lnstitut Henri Poincaré - Probabilités et Statistiques

and the transition

probabilities

^{for the} environment at the site x E Z" are

c ( . , . )

^and

q ( . , . ) satisfy

^some

compatibility conditions,

^{and e is a small}

parameter.

Under some

general assumptions

^we deduced in the paper

[2]

^{the local}

central limit theorem for the

displacements

^{of the}

particle.

^{In Part I}

we studied the

asymptotic (in time) decay

^{of the} correlations of the environment in a fixed frame of reference. The

present

paper is devoted to the

investigation

of the environment process in ^a frame of reference which

moves with the

particle, ^i.e.,

^{of the}

process {~t : t

^E

Z+}

^{defined as}

qt

(x) _ ~t (Xt

⁺

x) (sometimes

called "field from the

point

of view of the

particle").

~t evolves as an infinite-dimensional Markov chain.

We prove for any dimension v

that,

^{as t ~} oo, the distribution

of ~t

tends to a

limiting

invariant distribution _p, which is

absolutely

^continuous

with

respect

^{to the invariant} distribution

IIo

⁼ of the environment in the

unperturbed

^case

(e

⁼

0).

^{We also}

study

^{the time}

asymptotics

^{of the}

(time)

correlations of the field for

v &#x3E;

3. We prove that the

leading

term is of the

type

^const

e-at t- 2 .

The constant factor

depends

^{on the}

initial

conditions,

^{whereas a E}

(0, oo ) depends only

^{on the}

parameters

^of the model. This result should be

compared

^{with the}

long

time tail of the correlations of the field in a fixed frame of

reference,

^{which was studied in}

Part I. The different behavior is

explained by

^{the fact that in a} fixed frame of reference the environment

process 03BEt by

^{itself is not Markov.}

The methods used in the

proofs

^{are, as in the}

previous

papers

[1] ]

^and

[2],

^{based on the}

spectral analysis

^of the stochastic

operator (or

^transfer

matrix) T

of the Markov

chain {~t : t

^E

Z+}, acting

^on the space

=

L2 (H, IIo).

^{Here H = is the state} space of the field. The method of the

proof

^is

^based,

^{as in Part}

I,

^{on the}

analysis

^{of the}

leading spectral subspaces.

The paper is

organized

^as follows. Section 2 is devoted to the definition of the model and to the statement of the results. In Section 3 we prove the main technical theorem

(Theorem 3.1),

^which

gives

^the

decomposition

^{of J-l}

in invariant

(with respect

^to

T) subspaces.

Section 4 contain the

proofs

^of

the

theorems,

^which

rely

^on the results of Section

3,

and Section 5 is devoted to some

concluding

remarks. In the

Appendix

^we prove ^a technical lemma which is needed in the

proof

of Theorem 3.1.

Vol. 30, n° 4-1994.

562 C. BOLDRIGHIM, R. A. MINLOS AND A. PELLEGRINOTTI

2. DEFINITIONS AND FORMULATION OF THE RESULTS The model is described in detail in Section 2 of Part

I,

^{to which we} refer. We state here

only

^the

assumption

^{which differ from the ones of}

Part I.

Throughout

the paper ^we will write

(I n.m)

^to denote formula

(n.m)

of Part I.

The which

plays

^an

important

^{role in what}

follows,

^{is defined}

by (I 3.1 a).

Throughout

the paper ^{we assume} conditions

I,

II and III of Part I on

the random walk transition

probabilities,

and conditions IV and V on the transition

probabilities

of the random field. Condition VI on the

spectrum

of

Qo

^is

replaced by

^the

following

^one.

VI*.

If I S ]

&#x3E; 2 the

spectrum

^of

Qo

is such that

Condition VI* is the old condition VI

plus

^the

assumption

^{that the}

eigenvalue

^is

nondegenerate (hence real,

^since

complex eigenvalues

occur

only

ⁱⁿ

conjugate pairs).

Further,

^{we assume} Condition VII of Part I for the function c, but

replace

Condition VIII

by

^the

following (stronger)

^condition.

VIII*. The Fourier coefficients

of the inverse of the function

po (~) [eq. (I

^2.3

d)] satisfy

^the

inequality

Inequality (2.1 b) implies

the existence of a

spectral

gap

We ^now define the random field which will be studied in this paper, the

"environment from the

point

of view of the

particle".

This is the random field _qt E E

Z+ given by

the relation

Annales de l’Institut Henri Poincaré - Probabilités ^et Statistiques

For any bounded functional F on S2 we have for the conditional _average

[with respect

^{to the} distribution

(I 2.1)]

where

Py (. ~

^{is a}

product

^measure

qI

( s, s’ ) ,

^s,

^s’

^{E S}

being

the matrix elements of the matrix

Qi

^defined

in eq.

(I 2.5b).

^The conditional _average

(2.2b)

^{does not}

depend

^on

^Xt-i,

and we can consider it as an average ^over the Markov

field {

^1]t

(y) },

^with

transition

probabilities

The stochastic

operator

^or transfer matrix T of this field is

defined,

^as

usual, by

its action on the bounded functionals F on H :

We denote the average of ^a random variable

f

^with

respect

^to any

measure v

by ( f )",

^{and the} correlations

by ( f, g ~" _ ( f g ~" - ( f )" ~ 9 ) ~.

We fix an initial distribution II of the field _~, induced

by

^{some initial}

distribution II of the

field ç

^{for a} fixed initial

position

xo ⁼

Xo

^{of the}

random walk. II is

simply

^{the shift of}

fI by

xo.

Pn

denotes the distribution

on the _{space of}

trajectories

of the Markov field

{ r~t :

^{t E}

7L+ } generated by

the initial distribution II. From the definition

(2.3)

^we

find,

^{for F as before}

We shall also consider condition IX of Part I

(symmetry

^{of the random}

walk).

^{For the}

field 17

^it

implies

^the

following

statement, which may be considered as a new condition.

Vol. 30, n° 4-1994.

564 C. BOLDRIGHINI, ^{R. A.} MINLOS AND A. PELLEGRINOTTI

IX*.

Here,

^{as in Part}

I,

^V is the space reflection:

(V ~7) (x) _ ~ (-:c).

We need the

following

^further

assumption

^{on the} correlations of the initial measure II.

X*. The

spatial

correlations with

respect

^to the distribution II of the vectors

{03A80393:

^{r E of} the basis in H

[see (I 3.1a)] satisfy

^the

following

^cluster

inequality

Here denotes the multi-index with supp r

= ~ x ~

^~y

(x)

⁼

j,

^and

M, q

^{are two constants} such that M &#x3E;

1,

^and

q

^E

(0, 1).

As in Part

I, by dA,

^{A c we denote the minimal}

length

^{of the}

connected

graphs which join

^all

points

^{of the set A.}

We now formulate the main results of our paper. We understand

throughout

that condition I-IX are the ones of Part I. Conditions introduced in the

present

paper ^are denoted

by

^{a star.}

THEOREM 2.1. - Let _{t E}

l~+

^{denote the}

family o,f’measures generated by

^{the Markov} process r~t with ⁼

II,

^{and assume} conditions

I-V, VI*,

VII and VIII*.

Then,

^{as t ~} ~, the measures tend

weakly

^{to an}

invariant measure p, which is

absolutely

continuous with

respect

^{to the}

independent

^measure

no.

Moreover there are

positive

constants cl and

q

^E

(0, 1)

^{such that}

Remark 2.1. -

Inequality (2.4) implies

that Condition X* above holds for the invariant measure J1 ^as well.

In Theorem 2.1 Condition VI* could be

replaced by

the weaker Condition

VI,

^{at the cost of a}

longer proof.

We shall indicate below how it ^can be done.

The other result concerns the behavior of the time correlations of the

field,

Annales de l ’lnstitut Henri Poincaré - Probabilités et Statistiques

for which we write down the

precise asymptotics.

Consider the correlation

where _{x1, x2} are two fixed

points.

THEOREM 2.2. - Let

v &#x3E; 3,

^{and add} conditions IX* and X * to the

hypotheses of

Theorem 2.1.

Then,

^{as t --~} oo,

where ^a &#x3E; 0 is a constant

depending only

^{on the}

parameters of

^{the model}

and the constant C

depends

^{in addition on}

f 1, f 2 ,

^{x 1, x2, and II.}

By

^Remark 2.1 the invariant measure ~c satisfies the conditions of Theorem

2.2,

^{so that the}

equilibrium

^time correlations for the _{measure p}

are also of the

type (2.6).

3. EXISTENCE AND PROPERTIES OF THE INVARIANT SUBSPACES

3.1. More notation and

preliminary

^results

In what follows we denote

by

^const any absolute constant,

independent

of c.

The action

of T

on the basis

functions {03A80393 : 0393 ~ 203C0} is given by

Here

Aj (x)

r is the multi-index obtained

by replacing

^the

value, (x)

^{of r}

at x

with j, leaving

all the other

values, (~), ~ 7~ ~ unchanged. ^T°

^{is the}

unperturbed operator

Vol. 30, nO 4-1994.

566 C. BOLDRIGHINI, R. A. MINLOS AND A. PELLEGRINOTTI

and the coefficients

L ( ... )

^are

given by

Here we have used the

equality Aj ( - y )

^F

^{+ y}

⁼

Aj (0) (F

⁺

y ) ,

^{and the}

coefficients cj

(y), qj’j ,

^and

bjm,

^{m’ are the} coefficients of the

expansions

ⁱⁿ

the

basis (

^{of the functions c}

(u, s), 03A3 ^{q (s, s’)}

^ej

^(s’)

^{and ek}

(s) em (s) , respectively [see (I 2.6f)].

^{Note that} co

(u)

^{= 0}

and q0k

⁼

^0,

^which

implies L(0, j; u)

⁼ Cj

{u) .

We shall write T ⁼

T°

is translation invariant whereas T is not.

LEMMA 3.1. - T and 0 are bounded linear

operators

^{on ?-~.}

Proof. - By

eqs.

(3.1a, b, c)

^{we have}

where we use the notation

Recalling

^the

expression (I 3.1b)

of the scalar

product

ⁱⁿ

H,

^{we have}

Setting,

^{for fixed} y, F* ⁼

T"

_{+ y,} ⁼

f r~ _y,

^{we have}

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

where the

operator

^{is defined}

by

^the

position

The

proof

^that

^{T (y)}

^{is a bounded}

operator

^{is done}

exactly

^{as in Lemma A.I}

of

[2],

^and

since II by

translation invariance of the _norm, and the sum is over the finite

set I y ] ^D,

^{it follows that T is a bounded}

operator

^{in H. The}

proof

^{that A is bounded is an} easy consequence.

Lemma 3.1 is

proved..

As in Part I we

consider,

^{for M} &#x3E; M* -

max {max|ej (s)|, 2 },

^the

s

dense

subspace ^HM

^C

^~-l,

^with

norm ] ] ~~M

^defined

by (I 3.5a).

By

the definition

(I 3.1a)

^{of the basis}

{ ~r }

it is easy ^{to see} that the

following inequalities

^{between norms hold}

where II

^m

sup ~/ (~) ~ ]

^denotes the supremum ^norm

of f.

For the space

HM

^{we state} the

analogue

of Lemma 3.1.

LEMMA 3.2. -

(i)

^The

operators

^{T and 0394 are bounded on}

HM . (ii) HM

is invariant under T.

Proof. -

^{It is the same as the}

proof

^{of the}

analogous

Lemma 3.4 in

[2] ..

We denote

by

^{the set of the}

equivalence

classes of multi-indices which differ

only by

^{a shift.}

Let (

^{E and}

F _{E (be}

a

representative

of the class

(.

^Since

^T°

is translation

invariant,

^the

subspace consisting

of the vectors

is invariant under It is _easy to see that

Hence the

spectrum

^{of the}

operator x ~

^coincides with the range of the function Note that does not

depend

^{on the choice of}

the

representative r

E

(.

^{We denote}

by x°

the space

~C~ when (

^{is the}

class of

equivalence

^{of the} multi-indices F which contains

(defined

ⁱⁿ

Vol. 30, n° ^4-1994.

568 C. BOLDRIGHINI, R. A. MINLOS AND A. PELLEGRINOTTI

Condition X*

of § 2).

^{Condition VIII*}

implies

^{that the}

spectrum

^of

T°

in the space

~C°

^is

separated

^{from the}

spectrum

^of

2~°

in

~-C B ?-~o .

3.2. Construction of the invariant

subspaces

The main result of this section is the

following

^theorem.

THEOREM 3.1. - Under the

assumptions ^{I-V, VI*, VII,}

^and

^VIII*,

^and

for ~

^small

enough

^the

following

assertions hold.

(i)

^{One can}

find

^two

subspaces of ~-C, ~1

^and

^~nC2,

^invariant with respect

to the

operator T,

^{such that}

being

the space

of

^{the constants.}

(ii)

^{In the} space ^{one can}

find

^a

basis ~ hz ,

^{z E on which the}

operator T1 ~ T

^acts

according

^{to the}

formula

Moreover

p (y)

^{is real and}

^{given by}

the relation

and, for

^some q ^E

(0, 1),

^the

following inequalities

^hold

(iii)

^{The norms}

of

^the restrictions

of

^{T to the}

subspaces

^and

(endowed

^{with the}

norm ~ ~ . ~ satisfy respectively

the estimates

iv)

Under the additional condition IX*

p (y)

^{is an even}

function of

y E ~".

Remark 3.1. - The

decomposition ^{(I 3.2)}

^{of the}

space ~

^{in a direct}

integral

^{of the}

eigenspaces

of the group of

translations {Uv : v

does not reduce the

operator T,

since for c &#x3E; 0 T does not commute with

Annales de l’lnstitut Henri Poincaré - Probabilités ^et Statistiques

the space shifts. This is

why

^an additional sum appears in

(3.4a),

^{and we}

had to assume

inequality (2.1c)

^{in Condition VIII*.}

Remark 3.2. - If we have condition

VI,

instead of

VI*, i.e.,

^the

eigenvalue ~ 1

^{has a}

multiplicity s

&#x3E;

1,

Theorem 3.1 holds with some

obvious

changes. Namely

^{the basis in} the space will be labeled

by

two E

ll v , i = 1, ... , s ~ .

^{Moreover the}

operator T |H(1)M ~ ^T1

ⁱⁿ this basis will act as follows

M

where

and the matrix

elements {03B4ij (u)}, and {Sij (z, y)} satisfy

^estimates

analogous

^to

^(3.4c).

Proof of

^{Theorem 3.1. - The}

proof

^{makes use of the same ideas as the}

analogous proof

^{in section 3 of}

^[2].

^{It is based on several}

^lemmas,

^which

are stated and

proved

^{in the rest of the}

present

^section.

We shall first find the invariant

subspace

^{and then obtain}

Hi

^as

the closure of in H.

For c ⁼ 0 we have

clearly

~C = ~-lo ~-- ~C° + ~‘~C2 ,

~i, ^{M +} ~2 ~

^M,

~° l,,l = ?-~C° n ~-~CM 2 = 1,

^2.

Here

Ho

^is the space of the constants,

?-~°

^{was defined above and}

Hg

^is

the

(closed) subspace spanned by

^the

functions {03A80393:

^{1 r E}

g2}, g2 being

the set of the multi-indices r with

either supp h ~ ]

&#x3E; 1

or supp r ~ [

^{= 1}

but 03B3 (x) ~

^{1 for x E supp r.}

It is convenient for the moment to set

so that

HM

⁼

H01,

^{M + M, and T}

^(as

^an

^operator

^on is written

as an

operator

^matrix

’

Vol. 30, nO ^4-1994.

570 C. BOLDRIGHINI, R. A. MINLOS AND A. PELLEGRINOTTI

where the

operators Ti~

^{act as follows:}

^{Tll :}

~ -~ ~,

,

~21 :

~

-~ ~ l°~M, ^{T22 :}

^The

space , is

spanned by

^{the . : x E}

and 03A80.

i

We first

study

the inverse of the

operator T~l

in the space l,,l.

is defined in Condition X* of Section

2.)

We introduce for

brevity

^the

notation _~pz

= ~ riz ~ ,

^{and we will allow z to take the value}

^{0 :}

^{cpo =}

^Wj.

For the indices of the matrix elements of

operators

^on

~l°

^we shall also write z instead of

LEMMA 3.3. - The

operator T11

^{= T}

~~a

^is invertible in and

1, ^{M ,}

its inverse is

given by

,

where the

operator

^{D has the}

following properties:

for

^{some 0 E}

(0, 1).

Proof. -

^The

operator 7n

^can be written as

and,

^as it follows from formulas

(3.1 a, b, c) 7n

^{can be written as a matrix}

(corresponding

^{to the}

decomposition

⁼

^Ho

⁺

~C°, ^{~ )}

where

Its inverse is

represented by

the matrix

with

Annales de l’lnstitut Henri Poincaré - Probabilités ^et Statistiques

and R is

given by

^{a series}

where in the

running

^{term of the sum the}

product R

is

repeated k

times. From

eq. (3.8d)

^we

get

^const

^{Bf x-~ ~}

^{I for some}

0 E

(0, 1). Inserting

^the

expression

^for

7Z _{given by (3.8b)}

one

easily

finds the estimate

One finds also that

This _proves Lemma 3.3..

We look for an invariant

subspace

^{of the form}

where S : --~

?nC°, ~

^{is an unknown}

^operator.

The invariance condition for leads to

the following equation

^{for S}

[which

^is

analogous to eq. (I 3.6c)] :

JC is considered as

acting

^on the space

A (?-L°, ~ , ?nC°, ~ )

^{of the maps from}

to

?nC°,

^{~, endowed with the}

^operator

^norm.

^{Equation (3.9b)}

^{has a}

unique solution,

^{as stated}

by

^the

following

^lemma.

LEMMA 3.4. -

If ~

^{is small}

enough

^{one can}

find

^{a number} K0 &#x3E; 0 such that the _map lC is ^a contraction in the

sphere of

^radius ~o centered at the

origin of

^the space

A (~C°

^M,

~C° ~ ) .

Proof -

We denote with

Sz,

^r the matrix elements of the

operator

^S.

By equations b, c)

^{we see that}

(~21 )o, r

⁼

^0,

^{and for z E 7~v}

For

(T12 ) r, z

^we

^have,

^if

supp r = ~ ~c ~,

Vol. 30, n° 4-1994.

572 C. BOLDRIGHINI, ^{R. A. MINLOS AND A.} PELLEGRINOTTI

and if supp r

= ~ u ~

^U

{ v },

^{and z E Z"}

= 0 in all other cases.

For the norms of the

operators appearing

ⁱⁿ

equation (3.9b)

^{we have}

where the constants

Ci, j, ^{i, j}

⁼

^1,

²

^depend

^{on the}

parameters

^{of the model} and on M.

(3.12a)

^{comes from Lemma}

3.3, by observing

^that

Inequalities (3.12b, c)

^are

easily derived, using

^the

explicit expressions (3.10), (3.1 la, b)

^and

(3.12d)

^follows

by observing

that for ~ ⁼

0, ?22

with ( T° ~ ~ _

^{~c*, and that}

~-1 (~22 -

^{is a bounded}

operator (Lemma 3.2).

By inequalities (3.12a, b,

c,

d)

^{and Condition VIII*}

[inequality (2.1b)],

we see that the second term on the

right-hand

side of eq.

(3.9b)

^is

bounded,

for small _c,

by ~3 ( ( S ( ~,

^where

/3

^E

(0, 1).

Since the other two terms are

of order c

and ~ ~S~ 112 respectively,

^{K is a} contraction in any

sufficiently

small

sphere

^of Lemma 3.4 is

proved..

Hence there is a

unique S,

solution of eq.

(3.9b)

and the space defined

by

eq.

(3.9a)

is invariant with

respect

^{to T. Since M} &#x3E;

1,

it is not difficult to see that

Annales de l ’lnstitut Henri Poincaré - Probabilités et Statistiques

The Banach _space of the elements of

A (~‘~C°,

^~,

~C2, ^~)

^{with the norm}

( M

is denoted

by AM.

The solution S

of equation ^(3.9b)

^is

^actually

in

AM,

^{as it follows from the}

following

^lemma.

LEMMA 3.5. - The

right-hand

^side

of equation ^(3.9b) defines

^a map

AM ~ AM,

^and

for ~

^small

enough

^one can

find

^a

positive

^{number R}

so small that

J’CM

^{is a} contraction in the

sphere of

radius ~ centered at the

origin of AM.

Proof. -

^We

have, by inequalities (3.12a, b,

c,

d)

For the second term, ^{which is} the

leading

^{one, we have}

where

/3

^E

(0, 1 ) .

The conclusion follows as for the

preceding

^lemma..

Hence there is a

unique

^solution of eq.

(3.9b),

^{which coincides}

with the

previous

one. ^, We introduce a basis in

by setting

The action of T on the new basis is

given by

^the

following

^formulas

Vol. 30, nO ^4-1994.

574 C. BOLDRIGHINI, R. A. MINLOS AND A. PELLEGRINOTTI

We need to prove ^a

property

^{of fast}

decay

^as z,

z’

^{-~ o0} of the matrix elements

Az,

^{z~ . To this}

^aim,

^{we use the fact}

^that,

^{as it was}

^proved

ⁱⁿ

^[3],

S can be written as a series

where the summation goes ^over all sequences of

pairs

^{ai =}

(si, qj),

Si, qi E

Z+,

and x~l, ", , ar ^are real numbers

(see [3]).

^We first prove the convergence of the series.

LEMMA 3.6. - For ~ small

enough

the series

(3.16a)

converges in the

norm

of AM.

Proof. - By reasoning exactly

^{as in the deduction of}

inequality ^(3.13a)

we find

and,

ⁱⁿ

analogy

^with the deduction of

inequality ^(3.13b),

^{we find}

where,

^as

before, by [ [ . [ we

denote the

operator

^{norm of}

722 :

~(o~

’

2, M.

Now, by

^Lemma

3.3, expanding

the power

(Tl 1 ) -S

⁼

( (~° ) - ~ -f- ~ D) s,

we find

where

and

D(q)

is written as a sum of

products.

^The

^function

^{which is}

the Fourier transform of

(T011)-1,

^is

analytic

^{in some}

complex neighborhood Wd

⁼

{ A : ~ d,

^{z =}

1,... ,~ }

^{of the torus}

^T",

^{if d is}

sufficientely small,

and satisfies in

Wd

^the

inequality

where r

( ~c) , u

^{E Z" are} the Fourier coefficients

(2. la),

^{and the}

quantity

r~

is small for small d. It is _easy to see that the Fourier transforms

are

analytic

^for

^~,

^{~’ E}

^{Wd. Using}

^the

inequality (1

⁺

x)s -

¹

x 2

( 1

⁺

x 2 ) s ,

^{we find}

where _{ci, c2} are constants

independent

^{of s.}

^{[Note that, by}

^formula

^(3.8c)

we have

( ~11 ) oo

^{= 1 and}

^{= 0, z}

^{E Z" U}

^{0.] ]}

^In

computing

^the

Fourier coefficients we can now shift the

integration region

^{in the}

complex region Wd,

^{and we}

_get

^from

(3.19), (3.20a, b) (maybe by redefining r~)

the relations

Relations

(3.21) imply

..u C L- ^{.. - ..}

where c is a constant

independent

^{of s. From}

(3.17), (3.12d), (3.22)

^{we find}

Vol. 30, nO ^4-1994.

576 C. BOLDRIGHINI, R. A. MINLOS AND A. PELLEGRINOTTI

whence

r r

where Sr

⁼

Y~

^{Si and}

^Qr

⁼

1 1

From the

analysis

in paper

[3]

^{it follows that} xai, ... , ~r = ~ if and that

where ws, r ^{are the} coefficients of the

expansion

of the solution w

(z, ()

^{of the}

equation

By looking

^{at the}

singularities

^{in the}

complex z-plane

of the solution of the

quadratic equation (3.24b)

^{it is not hard to see} that the radius of convergence of the power series in ^z of w

((, z)

^{tends to 1}

as (

^{--~ 0. Since}

in our

case ~ ~ ~ _ (C ~)2,

and the number

1

^{+ ~}

C22 ) ^{a + ~ is,} by

Condition

VIII*,

less than I for e and d small

enough, B conclude

^that

Condition

VIII*,

less than 1 for ~ and d small

enough,

^we conclude that there is a

positive

^number 6-0 such that

for, s

co,

(1+~ C22 ) (a + ~ |1|)

will be smaller than the radius of convergence

(in z)

^{of the series}

(3.24a),

^which

implies

^absolute convergence in

AM

^{for the series}

(3.16a).

Lemma 3.6 is

proved..

Remark 3.3. - Note that since and

T21 03A80

⁼

^0,

^it

follows that

0,

^which

implies

~co ⁼

~o~

Remark 3.4. -

Inequality (3.23b) implies [

^{const ~.}

For what follows we need a more accurate

analysis

of the matrix elements of the

operator

^{S. What we need is}

provided by

^the

following

^Lemma.

LEMMA 3.7. - The

following

relations hold

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

where the

functions

^{R and V}

satisfy, for

^{some 8 E}

(0, 1),

the estimates

Proof. -

^The

proof

^{is rather}

lengthy

^and

requires

additional constructions.

It is deferred to the

Appendix..

By

Lemmas 3.5 and 3.7 we have

proved

^that is invariant with

respect

^to

T,

^{and that one can} find in it a

basis {uz : z

^{E Z" U}

0}

^on

which T acts as in

(3.15a),

^{and the}

coefficients ~ Az, z~ : z, z’

^{E Z" U}

0 ~

are,

by (3.8a, b), (3.15b)

^and

(3.25a),

To

accomplish

^the

proof

^{of assertion}

(ii)

^of Theorem 3.1 we introduce in a new basis

and we show that the

sequence H

⁼

f Hz : z

^E

7L" }

^can be determined in such a way that

In

fact, by

eqs.

(3.15a)

^and

(3.27a)

^{we have}

and we can define H to be the solution of the

system

^of

equations

Let

Bq, q

^E

(0, 1)

be the space of the sequences ^H such that the norm

is finite. From formula

(3.15b),

^Lemmas 3.3 and 3.7 it follows that for q &#x3E; 6~ the z’ E defines a bounded

operator A

in

Bq,

^with

1, and, moreover {Az,0 :

^{z E E}

Bq.

Vol. 30, n° 4-1994.

578 C. BOLDRIGHINI, R. A. MINLOS AND A. PELLEGRINOTTI

Hence there is a

unique

^{solution in}

Bq

^{of the}

system (3.28a). Moreover, by

^formula

(3.26)

^{it is clear}

II q ~ ^const

whence it follows that the same

inequality

holds for the solution of _eq.

(3.28a).

By (3.27b)

the space

splits

^{into two invariant}

subspaces:

^the

subspace

^{of the constants}

1-lo

⁼

{

^c

ho } _ ~

^c

~o }.and

^the

subspace

which is

spanned by

^the

basis {hz : z

^E

7L" }. Setting

^now

b (y)

^{= R}

(y),

and

S (x, y)

⁼

y)

⁺

Vx,y, using

^{the estimates}

(3.25b)

^we

_get

^the

relations

(3.4a, b, c)

of Theorem 3.1.

We now prove

inequality (3.5a)

^{for the} restriction of T to

Using

^the

explicit expression (3.27a)

^{for the functions of the}

basis { ^h,z },

^{Remark 3.4}

and

inequality ^(3.28b)

^{we see that}

where the constant is

independent

^{of ~ and z.} For any ^{vector v =}

LV z h z

^E

z

~CM , by susbstituting

^the

explicit expression (3.27a)

^{of the functions}

hz,

we find

whence, reasoning

^as

above,

^one

deduces,

^{for some} ci &#x3E;

0,

Using (3.29), (3.30)

^{and the}

inequality

H ‘

which follows

easily

^from

equation (3.26),

and Lemmas 3.3 and

3.7,

^we

find,

for ~ small

enough

Annales de l ’lnstitut Henri Poincaré - Probabilités et Statistiques

Inequality (3.5a)

^is

proved.

We are left with the construction of the space

7-lM~.

^{We recall that in}

^[2]

we introduced a basis

{ 03A8*0393 :

^{r E in the}

^{space H}

=

0, ... , |S| - 1 }

^{is a} basis in the space

l2 (8, given by

the normalized

eigenvectors

^{of the}

operator

^and

bi-orthogonal

to the

basis {ej : j

⁼

0, ..., |S| - 1}.

^{The r E is}

bi-orthogonal

^{to the}

basis {03A80393

¹

^{r E}

in H. We observe

furthermore

that the

matrix {T*0393.0393’} of

^the

^{operator T*, adjoint of T,}

in the basis r E is

the _adjoint

of the

matrix { 7r, r~ ~

^{of the}

operator

^{T in} the

basis {03A80393,

^{r E}

^i.e.,

By applying

^{to the}

operator

^{T* the same} considerations as above we find

a

subspace

^C

1-lM,

invariant with

respect

^to

T*, which,

ⁱⁿ

analogy

with

(3.9a),

^is

given by

Here is the _space

spanned by

^{the vectors}

~po

^and

{ cpz

^:

z E

cpz = ~ ~~ z ~ ,

^{is defined in}

^analogy

^{to and the}

^operator

s* : is the _space

spanned by

^{the vectors}

~ ~ r :

^{r E}

~2 ~ )

^is defined in

analogy

^{to S. In the} space ^{we can} choose a E Z" U

0 ~

of the form

~cz

⁼⁼

cp;

^{+ ?* It is}

not hard to see that

T* acts on the basis

{ ~cz ~

^{as follows}

and,

ⁱⁿ

analogy

^with eq.s

(3.26),

^{= 1 and =} 0 for all z E Z".

As above we can go ^{over to a new} basis in

and the

sequence {H*z, z

^{E is the solution of the}

system

Vol. 30, n° 4-1994.

580 C. BOLDRIGHINI, ^{R. A.} MINLOS AND A. PELLEGRINOTTI

which, reasoning

^as for eq.

(3.28a),

^is in the space

Bg.

^We

^find,

^as

^above,

i. e.,

^the space

* splits

^{into two invariant}

(with respect

^to

T* ) subspaces:

and ^*

being spanned by

^{the vectors}

ho,

We denote

by

^{and the}

subspaces

of ~-C obtained

by taking

the closure of the _spaces ^* and

7~

^{* in the norm of ~.}

^They

^are

invariant with

respect

^{to T* and for them a}

decomposition analogous

^to

(3.35)

^holds.

We now come to the

proof

^{of the}

reality

^of the function

p (~/).

^{It follows}

from the

reality

^{of the}

operator

^T

(i.e.,

^{from the fact that}

T f

⁼

T f ),

^and

the

reality

^{of the functions}

( ~ )

^{= ei}

( ~ ( z ) ) : z

^{E and}

which,

in its turn, follows from Condition VIII*. This

implies

^the

reality

^{of the}

operators 7n, 7i2, T21,

^and

?22.

In

fact,

consider for instance a real function

f

^E

?-~°,

^{M . As T is}

^real,

T

f

⁼

?i2 f

⁺

?22 f

^is

^real,

^{and so is}

T~12 f = L ^{(T f, cpz)}

^~pz.

Hence

7i2

^and

722

^{are real}

operators. Analogous arguments

^{show that}

7n

and

Tl2

^are also real. This

implies, by equation (3.16a),

^the

reality of S,

and hence the

reality

^{of the}

basis {hz : z

^E and of the matrix elements

Az,

^{z~ .}

^Since,

^as it follows from estimate

(3.4c)

^{and formulas}

(3.26), p (y)

^{= lim}

Az,

^{z-y, is a real function.}

If we now assume that the

symmetry

^{Condition IX*}

holds,

^{it is not hard}

to see that

Az, z~

⁼

Az~,

^{z. In}

^fact, by

the discussion of Condition IX in Part

I, Po ( y )

^and cj

(y), j

⁼

1, ... ~ S ~ -

^{1 are}

symmetric in y

^{E It}

follows that the

operators 0~

defined

by

^formula

(A.4)

^{of the}

Appendix,

are also

symmetric

ⁱⁿ yj. ^This

gives

^the

required property

of the matrix

Az,

^z~,

which,

^{with the}

help

^of

(3.26)

^{leads to the}

symmetry

of the function

p ( y) .

Assertions

(ii)

^and

(iv)

of Theorem 3.1 are

proved.

We now pass ^to the rest of the

proof.

^{We have}

As a consequence of the

bi-orthogonality

^{of the}

bases {

^{r e}

m}

and {Wr :

^{r E}

hj

^is

orthogonal

^{to the}

subspace ?-~1.

^Moreover

ho

is

orthogonal

^{to the}

subspace

^In

fact, by

^the invariance

of h*

^with

respect

^{to T* we have}

( ho , (E - T) v )

⁼ 0 for any v ^E

?-~~ ,

^{where E}

denotes the

identity operator. But, by inequality (3.5a) f - ~

^{is invertible}

Annales de l ’lnstitut Henri Poincaré - Probabilités et Statistiques

in

~‘~CM ,

^which

^implies

^that

ho

^{1 v for any v E}

~M .

^{Since is dense}

in

Hi,

^the assertion follows.

Next we construct the

bi-orthogonal

^basis

to {hz : z

^E

7~v ~. By

relations

(3.14), (3.26), (3.31)

^and

(3.32)

^{we have}

We introduce a new E in the _space of the form

The matrix

.~ _ ~ .~’z, z~ ~

^{is chosen in such a} way that

where 7* is the

adjoint

matrix of .~’.

In Lemma 3.8 below we prove that the

operator

^{C has a small norm} in

Bq .

^Hence

^.~’,

^{which can} be written as

exists as an

operator

^on

Bq,

and has also small norm.

An easy check shows that the ^E is

bi-orthogonal

to the

basis {hz :

^{z E in}

^xl.

We now introduce _{the space}

Clearly

this space is invariant with

respect

^to

T,

^{and for} any ^vector

f

we have a

unique decomposition

’

This last assertion can be

proved

^{as follows. We set}

It is easy ^{to see} that

f(2)

¹

ho

^and

f(2)

¹

?~Ci,

^{so that}

f(2)

^E

H2.

^The

uniqueness

^{of the}

decomposition (3.37a)

^{is evident.}

This proves the

decomposition (3.3a).

^To

accomplish

^the

proof

^we

have to prove the

analogous

^assertion

(3.3b)

^for

HM.

^{We have to} show

Vol. 30, n° 4-1994.

582 C. BOLDRIGHINI, R. A. MINLOS AND A. PELLEGRINOTTI

that and

f t2~

^{are in and}

~C~ , respectively,

^{for any}

f

^E

HM.

^We

first need an estimate for the matrix elements

Sz, ^rand Sz

^r.

LEMMA 3.8. - The

following

^{estimates hold}

where _{q E}

(0, 1 ) , ^Cl

^{is an absolute} constant, and ^u is the element

of

^the

set supp r which is closest to z. A similar estimate holds

for Sz,

^{r, z E}

^7~v,

and

finally

^"

Proof. -

^{It is not hard to} deduce the estimate

(3.38a) by analyzing

^the

terms of the series

(3.16a, b),

^{much in the same} way ^as it is was done in the

proof

^of Lemma 3.7. As for

S*,

^{it can be}

represented

^{in a}

^series,

^which

is

simply

^obtained

by replacing

^{in the series}

(3.16a, b)

the matrix elements of the

operator

^{T with the} elements of the

operator T*.

^The estimates that

are needed are obtained as for S. We omit the details..

To _prove that the

projection

^of any ^vector

f

^E

^HM

^on

^H2

^and

Hi belongs respectively

^{to or}

~nC~ ,

^and

inequality (3.5b)

^{we introduce}

a basis in

H2 .

Consider the functions

_ - - -

which,

^{as it} is easy ^to

check, belong

^{to the} space

H2.

^{In addition we set}

= uz, ^z e Z"

[see (3.14)],

^and

Uj

⁼ uj ⁼

Wj.

i

LEMMA 3.9. - The

following

assertions hold.

(I)

^The

functions ( Ur,

^{r e}

C2 )

^{are in}

H2,

^M.

(ii) Any

^vector

f

^e

H M

^{is written as a series}

which converges in the

and, for ~

^small

enough,

^{we have}

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques